Tunable quantum gaps to decouple carrier and phonon transport leading to high-performance thermoelectrics

Thermoelectrics enable direct heat-to-electricity transformation, but their performance has so far been restricted by the closely coupled carrier and phonon transport. Here, we demonstrate that the quantum gaps, a class of planar defects characterized by nano-sized potential wells, can decouple carrier and phonon transport by selectively scattering phonons while allowing carriers to pass effectively. We choose the van der Waals gap in GeTe-based materials as a representative example of the quantum gap to illustrate the decoupling mechanism. The nano-sized potential well of the quantum gap in GeTe-based materials is directly visualized by in situ electron holography. Moreover, a more diffused distribution of quantum gaps results in further reduction of lattice thermal conductivity, which leads to a peak ZT of 2.6 at 673 K and an average ZT of 1.6 (323–723 K) in a GeTe system. The quantum gap can also be engineered into other thermoelectrics, which provides a general method for boosting their thermoelectric performance.


Supplementary Discussion
The distribution of Bi The QGs stem from ordered Ge vacancies 1 , which are mainly introduced by Bi2Te3 in this work: The Bi elements may segregate near some QGs due to defect reaction, which will show a brighter contrast in the HAADF image (the QG labeled as N in Supplementary Fig.   1a). But not all the QGs (the QG labeled as M) have this kind of brighter contrast ( Supplementary Fig. 1a.) The missing layer of Ge causes the darker contrast of Ge mapping in Supplementary Fig. 1b Fig. 1d and e). The EDS spectrum ( Supplementary Fig. 1f) proves that all the above mappings are based on signals from strong peaks.
The Bi elements near QGs may be mildly segregated but not fully ordered. The amount of Bi doping and the vacancies should have a rough relationship of 2:1 according to eq. S1. If the Bi atoms are ordered near QGs, one layer of QG will cause the ordering of two-layer Bi (two sides of QGs), which will result in a sharp drop to almost zero of Bi content away from the QG, but the line scan across the QG indicates that Bi elements are almost constant away from the QG ( Supplementary Fig. 1e).
In summary, the Bi may have mild segregation near some QGs but not completely ordered.
Technical detail of DPC Differential Phase Contrast (DPC) imaging is used in STEM to directly measure local electromagnetic fields in materials. The converged electron beam deflected by the electric field will generate a differential signal which is recorded by the segmented detector. By analyzing the differential signal, we can map the electric field distribution.

S5
The raw data in this work are shown in Supplementary Fig. 3.
To exclude the effect of scanning noise, we apply DPC to the same area from two orthogonal directions (Supplementary Fig. 4a and b). We have also mapped six random areas ( Supplementary Fig. 4c-h), and the same result is obtained.
Technical detail of (in-situ) Electron Holography The original data for potential mapping of Ge0.867Re0.003Bi0.087Te are shown in Six potential mappings are obtained in Ge0.927Bi0.049Te ( Supplementary Fig. 7) using the same methods. The highly repeatable data prove that the results of quantum wells at QG are reliable. We obtain the same results of quantum well in two QG containing GeTe materials. This means that the QGs in different Ge-Bi-Te compounds have the same potentials.
The relationship between the dipoles and potential well The electric dipoles as observed by DPC imaging are caused by macroscopic Efield (Emacro), which is the gradient of the potential well ( Supplementary Fig. 8). In GeTe lattice without QG, the positively charged Ge layer and negatively charged Te layer stack alternatively, such that the E-field from Ge layer to Te layer is canceled by the E-field from Te layer to Ge layer ( Supplementary Fig. 8a). In this way, there is no net E-field on the macroscopic scale, which lowers the total electrostatic potential energy. However, the QG breaks the balance and net macro-E-field appears where Elocal, Emacro and are the local E-field, macro-E-field, and relative dielectric constant. The of GeTe is about 36 to 58 3,4 . As a result, Elocal is at least 13 times stronger than the Emacro. Thus, the observed E-field from the DPC is mainly from Elocal by dipoles ( Supplementary Fig. 8c).
Calculation of exchange-correlation potential.
The exchange-correlation potential VXC can be calculated by 5 where * is the effective Bohr radius for effective electron mass, is the dielectric constant, and n is the carrier concentration. We plot the calculated dependence on carrier concentration in and find the magnitude of is at the level of 10 -19 V which is far lower than the magnitude of Coulombic potential at the level of 8V (Fig. 2e).
Scattering problem of arbitrary potential.
The conduction electrons in the semiconductor can be well described by following where ℏ is the reduced Planck constant, m is the effective mass, k is the wave-vector, and U is the scattering potential. For arbitrary U, it is impossible to analytically solve this problem and numerical method is used. Firstly, by using the discrete Fourier transformation we get the tight-binding Hamiltonian in the real space,   Fig. 9). The above analysis is consistent with the experimental results. After the addition of QG, the DOS effective mass keeps almost constant ( Supplementary Fig. 13). Therefore, the lower Seebeck coefficients seen in Q group samples are mainly due to the higher carrier concentration than N group samples.

Calculation of lattice thermal conductivity
According to phonon Boltzmann transport theory, the lattice thermal conductivity S8 can be expressed as 11 where , Ω, , , , and are the total number of phonon states , volume, phonon distribution function, angular frequency, scattering rates, and group velocity, respectively. The dominant parameters are and . For a low , the desired should be large while the should be small.

Sound velocity and Grüneisen parameter
The average sound velocity (a) is calculated from the sound velocity as 12 .
where l is the longitudinal sound velocity and t is the transverse sound velocity.

Calculation of lattice thermal conductivity
The following formula is frequently used to calculate the Lorentz constant and the electron thermal conductivity 14-16 .
= 1.5 + exp − | | , (eq. S12) = − = − . (eq. S13) However, this is not accurate enough when the phonon scattering mechanism is not dominant. The accuracy can be improved according to the following formula.
For the Ge0.870Bi0.087Te, the QGs are formed by homogeneous nucleation, thus their spacings are random distributed with a peak at around 7 nm, while for Re-doped GeTe the Re-dopants produce a strong strain field due to the huge atomic radius difference between Re and Ge, (Re:188pm, Ge:125pm) 17 , which facilitates the inhomogeneous nucleation of QG and affect its distribution. The result is that the distribution is broadened. The Bi dopants can not play such a strong role as inhomogeneous nucleation center because of its similar atomic radius (143 pm) 10 .
Estimation of hole mean-free-path. 1% 0.5% 3% 1.5% 5% 2.5% 7% 3.5% These potential barriers can lead to the energy filter effect because low energy charge carriers have less transmission coefficient in contrast to high energy charge carriers.

Supplementary Tables
Supplementary Table 1